Magnetic field
Magnetic field
Ruslan Prozorov
5 October 2009
Physics 590B
popular definitions
• Magnetic
Magnetic fields surround magnetic materials and electric fields surround magnetic materials and electric
currents and are detected by the force they exert on other magnetic materials and moving electric charges. Th
The magnetic field at any given point is specified by both i fi ld
i
i i
ifi d b b h
a direction and a magnitude (or strength)
• magnetized region of space: a region of space magnetized region of space: a region of space
surrounding a magnetized body or current‐carrying circuit in which the resulting magnetic force can be detected
• A condition found in the region around a magnet or an electric current, characterized by the existence of a
electric current, characterized by the existence of a detectable magnetic force at every point in the region and by the existence of magnetic poles.
• A magnetic field is generated when electric charge A
ti fi ld i
t d h
l ti h
carriers such as electrons move through space or within an electrical conductor. history
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Earliest description of a magnetic field was performed by Petrus
p
g
p
y
Peregrinus
g
and published in his “Epistola
p
p
Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete” and is dated 1269 A.D. Petrus Peregrinus
mapped out the magnetic field on the surface of a spherical magnet. Noting that the resulting field lines crossed at two points he named those points 'poles' in analogy to Earth's poles. Three centuries later, near the end of the sixteenth century, William Gilbert of Colchester replicated Petrus
Peregrinus work and was the first to state explicitly that Earth itself was a magnet. William Gilbert's great work De Magnete was published in 1600 A.D. and helped to establish the study of magnetism as a science.
The modern distinction between the B‐ and H‐ fields was not needed until Siméon‐Denis Poisson (1781–1840) developed one of the first mathematical theories of magnetism. Poisson's model, developed in 1824, assumed that magnetism was due to magnetic charges. In analogy to electric charges, magnetic charges produce an H‐
field. In modern notation, Poisson's model is exactly analogous to electrostatics with the H‐field replacing the electric field E‐field and the B‐field replacing the auxiliary D‐field.
l
i fi ld E fi ld d h B fi ld
l i
h
ili
D fi ld
Hans Christian Oersted discovered that an electrical current generates a magnetic field that encircles the wire.
Andre Marie Ampere showed that parallel wires having currents in the same direction attract
Jean‐Baptiste Biot and Felix Savart developing the correct equation, the Biot‐Savart Law, for the magnetic field of y g
a current carrying wire.
In 1825, Ampere published his Ampere's Law which provided a more mathematically subtle and correct description of the magnetic field generated by a current than the Biot‐Savart Law.
Michael Faraday shows in 1831 that a changing magnetic field generates an encircling electric field.
In 1861, James Clerk‐Maxwell wrote a paper entitled 'On Physical Lines of Force' in which he attempted to explain Faraday'ss magnetic lines of force in terms of a sea of tiny molecular vortices. These molecular vortices occupied Faraday
magnetic lines of force in terms of a sea of tiny molecular vortices These molecular vortices occupied
all space and they were aligned in a solenoidal fashion such that their rotation axes traced out the magnetic lines of force. Although the classical theory of electrodynamics was essentially complete with Maxwell's equations, the twentieth century saw a number of improvements and extensions to the theory. Albert Einstein, in his great paper of 1905 that established relativity showed that both the electric and magnetic fields were part of the same
of 1905 that established relativity, showed that both the electric and magnetic fields were part of the same phenomena viewed from different reference frames. Finally, the emergent field of quantum mechanics was merged with electrodynamics to form quantum electrodynamics or QED.
Earth as a magnet
A sketch of Earth's magnetic field representing the source of Earth's magnetic field as a magnet. The north pole of earth is near the top of the diagram, the south pole near the bottom. Notice that the south pole of that magnet is deep in Earth's interior p
g
p
below Earth's North Magnetic Pole. Earth's magnetic field is produced in the outer liquid part of its core due to a dynamo that produce electrical currents there.
magnetic field due to current
cgs
SI
H (Oe)
H
(Oe)
B (G)
H (A/m)
H
(A/m)
B (T)
the right‐hand rule
The Maxwell’s Equations
in cgs units
original Maxwell’s equations
magnetic monopoles
magnetic vector potential
magnetic field momentum
Magnetic moment
Magnetic moment of a closed loop carrying current I:
Mi 
I
C r dl  ISn
2c 
Magnetic field on the axis of a loop of radius R at a distance z is:
Total magnetic moment: M   Mi
(superposition principle)
Hz 
2M i
 R2  z 2 
3/ 2
origin of magnetism (except for currents…)
M ion   J   g  B J
J  L  S
spin
orbital
total angular momentum
free electron:
g  2.0023  2.00
Magnetic moment:
g
M e  B
(J=S=1/2)
 ‐ gyromagnetic ratio
gg – Landé factor
e
erg
B 
 9.27410 1021
G
2mc
g  1
Bohr magneton
J  J  1  S  S  1  L  L  1
2 J  J  1
what about MPMS?
it measures a total magnetic moment in cgs (emu)
1 emu is:
1
emu is
• M of a 1 m2 loop carrying a 1 mA current
• M of a loop of radius 1.78 cm carrying a 1 A current
3) ~ 1 emu
• Typical permanent magnet (1 mm
l
(
)
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M of a neutron star ~ 1030 emu
The Earth’s magnetic moment ~ 8x1025 emu
An electron spin: B~10‐20 emu
Proton and neutron: 
N~10‐23 emu
• One Abrikosov vortex (0.1 mm long) ~ 10‐10 emu
• Change in M due to d
Change in M due to d‐wave
wave gap < 10
gap < 10‐10 emu/K
• Hard superconductors ~ 0.1 emu
magnetic moment in a magnetic field
torque:
Energy:
W  μB    B cos  
Force:
F   grad W   grad  μB 
τ  μB
B

for example, for

B   Bx  x  , 0, 0  , μ    x ,  y , 0 
μB   x Bx and
F  x
dBx
dx
in inhomogeneous magnetic field
x  0
x  0
B
F
B
F changes sign
h
i
however torque aligns along the field
per unit volume
Practical definitions
4 M  B  H
bound (molecular, spin etc) currents
f
free currents (moving charge)
t (
i
h
)
What is the problem with this definition?
What
is the problem with this definition?
1. Assumes uniform H (ellipsoids only)
2. Assumes uniform B (homogeneous system)
  M / H ‐ dimensionless!
 SI  4 cgs
some other quantities are used:

m 

 mol
 cc
1 
 cc  g  g 


 g
1 
 m M m 
 mol 

g
mol


B
H
x
Let’s see how well it works
a simple classical paramagnet
a simple classical paramagnet
M  M s L  x, x 
pB H
k BT
B
K 
 0.671
0 671  
kB
T 
0.4
1.0
0.3
dL/dx
x0.2
0.5
L(x))
L(x)
0.2
0.1
00
0.0
-0.5
0.0
-0.1
-1.0
-0.2
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
x=H/kBT
-60
-40
-20
0
x=H/kBT
20
40
60
Spatial inhomogeneity
M is not a single‐valued
M is not a single
valued function of H
function of H
demagnetization energy, magnetostatic energy, dipolar or g
gy p
fringe (stray) fields energy
~ M 2V
ferromagnetic domains
g
domain walls energy
 ~1
erg
cm 2
gain – energy of field outside
loose – energy of the domain wall
Magnetic hysteresis
type‐II superconductor
Meissner State
Partial Penetration
Trapped Flux
Using Faraday law of induction
1 d
V 
c dt
d
How to measure magnetic field?
Miniature Hall‐probe Array